On series of successes in Bernoulli sequences of random variables

Abstract

Some of the first studies in probability theory were associated with Bernoulli schemes - sequences of independent identically distributed random variables taking the values 1 with a certain probability 0 < p < 1 and 0 with a probability q = 1 - p. The study of sequences of such random variables led to the need to deal with a geometrically distributed random variable. Limit theorems for properly centered and normalized sums required the consideration and study of normally distributed random variables. Working with classical Bernoulli sequences and their other two-point generalizations led to the need to develop various methods for studying them, which were then used for other random variables. Despite the numerous results obtained since the publication of Jacob Bernoulli’s monograph "The Art of Conjectandi" in 1713, more and more new schemes for Bernoulli quantities are appearing that require further study. This work is a direct continuation of the authors’ articles published in 2022 and 2023.

Keywords:

Bernoulli scheme, binomial distribution, geometric distribution, expected value